Problem $4$: Practical ohmic contact

Consider a M$-$S junction formed by NiSi in contact with uniformly doped n$-$type Si $.$ The Schottky barrier height $\backslash (\backslash$Phi$\_\{\backslash$mathrm$\{$Bn$\}\}\backslash )$ is $0\mathrm{.}65$ eV $.$ This junction will pass high current under reverse bias $(\backslash ($ V$\_\{\backslash$mathrm$\{$A$\}\}<mn>0</mn><mtext></mtext><mi>\backslash </mi>$mathrm$\{$~V$\}\backslash ))$ by quantummechanical tunneling of electrons from the metal into the Si conduction band, if the barrier presented to the electrons is thin enough $\backslash ((<mn>1</mn><mn>0</mn><mtext></mtext><mi>\backslash </mi>$mathrm$\{$~nm$\})\backslash ).$ Assume that the onset of efficient tunneling occurs when the Fermi level in the metal is at the same level as the conduction band edge $\backslash ($ E$\_\{\backslash$mathrm$\{$c$\}\}\backslash )$ at a distance $10$ nm into the Si$.$

a$)$ What is the minimum value of $\backslash ($ N$\_\{\backslash$mathrm$\{$D$\}\}\backslash )$ such that this condition would be met in equilibrium?

$($Hint: Assume that the Fermi level is located at the conduction band edge, i$.$e$.\backslash ($ E$\_\{\backslash$mathrm$\{$F$\}\}=$E$\_\{\backslash$mathrm$\{$c$\}\}\backslash ),$ for very heavily doped n$-$type Si$.)$

b$)$ Sketch the energy band diagram under the conditions in part $($a$).$